Finite Group Databasebeta

← #26112823

Subgroups — #26112823

Order 12032.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254303C211yesmaximal, characteristic
2#20244228C211yesmaximal, characteristic, complemented
3#20242065C211yesmaximal, characteristic, complemented
4C2×#15688866C211yesmaximal, characteristic, complemented
5#20241285C211yesmaximal, characteristic, complemented
6#20241239C211yesmaximal, characteristic, complemented
7C47×#2935C211yesmaximal, characteristic, complemented
8C2×#13094021C2211yescharacteristic
9#15689499—21no—
10#15689471—21no—
11#15689318—22no—
12#15689256C414yes—
13#15689176C2211yescharacteristic
14#15689147C2211yescharacteristic
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×#13093940—21no—
17C2×#13093940—21no—
18#15688869—21no—
19#15688867—21no—
20#15688866—21no—
21#15688866—21no—
22C2×#131×C47—21no—
23C47×#532C2211yescharacteristic
24C47×#436—21no—
25C47×#435—22no—
26C8×D4×C47—21no—
27C2×C4⋊C8×C47C2211yescharacteristic
28C47×#408—21no—
29C4×M16×C47C2211yescharacteristic
30C22×D188—21no—
31C22×D188—21no—
32C22×D188C2×C412yes—
33C2×#10625334—24no—
34C2×#10625329—21no—
35C22×C47⋊C8—21no—
36#13094021—21no—
37#13094021—21no—
38#13094021—21no—
39#13094021—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45#13093944—24no—
46#13093940—42no—
47#13093940—42no—
48C22⋊(C47⋊C8)C2×C412yes—
49C4×C47⋊C8—21no—
50C2×#41×C47—21no—
51#131×C47—42no—
52#131×C47—42no—
53C2×M16×C47—22no—
54C2×M16×C47D411yescharacteristic
55C2×M16×C47D411yescharacteristic
56C22×C8×C47—22no—
57C22×C8×C47—21no—
58C4⋊Q8×C47D411yescharacteristic
59#127×C47D411yescharacteristic
60#124×C47—21no—
61#121×C47—21no—
62#121×C47C2×C412yes—
63C4×D4×C47—21no—
64C4×D4×C47D412yes—
65C2×C42×C47C2311yescharacteristic
66C4⋊C8×C47—21no—
67C4⋊C8×C47D411yescharacteristic
68C4⋊C8×C47D411yescharacteristic
69C22⋊C8×C47—22no—
70C22⋊C8×C47—22no—
71#97×C47—21no—
72C4×C8×C47—21no—
73C23×D47—22no—
74#41×C47—41no—
75#41×C47—41no—
76#41×C47—41no—
77#41×C47—41no—
78C2×Q8×C47—21no—
79C2×D4×C47—22no—
80C2×D4×C47—21no—
81C2×D4×C47—21no—
82C2×D4×C47C22⋊C412yes—
83C22×C4×C47—22no—
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47#4111yescharacteristic
87C2×D188—22no—
88C2×D188—22no—
89C2×D188—41no—
90C2×D188—41no—
91C2×D188—41no—
92C2×D188—41no—
93C2×D188—41no—
94C2×D188—41no—
95C2×D188—41no—
96C2×D188—41no—
97C2×D188—82no—
98C2×D188—81no—
99C2×D188—81no—
100M16×C47—21no—
101M16×C47—21no—
102M16×C47—21no—
103M16×C47—44no—
104M16×C47D812yescomplemented
105C2×C8×C47—21no—
106C2×C8×C47—21no—
107C2×C8×C47—21no—
108C2×C8×C47—21no—
109C2×C8×C47—44no—
110C2×C8×C47—41no—
111C2×C8×C47—41no—
112C4⋊C4×C47—21no—
113C4⋊C4×C47C22⋊C412yes—
114C22⋊C4×C47—22no—
115C22⋊C4×C47—22no—
116C42×C47C2×D411yescharacteristic
117C42×C47C2×D411yescharacteristic
118C42×C47C2×D411yescharacteristic
119C42×C47#4111yescharacteristic
120#10625334—84no—
121#10625329—41no—
122#10625329—41no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—81no—
127C23×C47—22no—
128C23×C47C4×D411yescharacteristic
129C22×D47—42no—
130C22×D47—42no—
131C22×D47—42no—
132C22×D47—82no—
133C22×D47—82no—
134Q8×C47—41no—
135Q8×C47—41no—
136D4×C47—42no—
137D4×C47—42no—
138D4×C47—42no—
139D4×C47—41no—
140D4×C47—41no—
141D4×C47—41no—
142D4×C47—41no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—22no—
146C2×C4×C47—41no—
147C2×C4×C47—42no—
148C2×C4×C47—42no—
149C2×C4×C47C2×C22⋊C411yescharacteristic
150C2×C4×C47C4×D411yescharacteristic
151C2×C4×C47#12011yescharacteristic
152C2×C4×C47#12111yescharacteristic
153C2×C4×C47#12111yescharacteristic
154C2×C4×C47#12311yescharacteristic
155C2×C4×C47C2×D811yescharacteristic
156C2×C4×C47#13511yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
159D188—44no—
160D188—82no—
161D188—82no—
162D188—81no—
163D188—81no—
164D188—81no—
165D188—81no—
166D188—81no—
167D188—81no—
168C8×C47—22no—
169C8×C47—41no—
170C8×C47—41no—
171C8×C47—41no—
172C47⋊C8—42no—
173C47⋊C8—81no—
174C47⋊C8—81no—
175#9273—471nomaximal, Sylow
176C22×C47—21no—
177C22×C47—21no—
178C22×C47—42no—
179C22×C47—42no—
180C22×C47—42no—
181C22×C47#38611yescharacteristic
182C22×C47C4×D811yescharacteristic
183C22×C47#44211yescharacteristic
184C2×D47—82no—
185C2×D47—82no—
186C2×D47—82no—
187C2×D47—162no—
188C2×D47—162no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—42no—
193C4×C47#44711yescharacteristic
194C4×C47#44911yescharacteristic
195C4×C47#45911yescharacteristic
196C4×C47#46311yescharacteristic
197C4×C47#47111yescharacteristic
198C4×C47#47211yescharacteristic
199#3205—471no—
200#2935D4711yescharacteristic, complemented
201#1965—471no—
202#1547—471no—
203C2×#331—471no—
204#1301—471no—
205#1283—471no—
206C2×C47—21no—
207C2×C47—21no—
208C2×C47—82no—
209C2×C47#187911yescharacteristic
210C2×C47#190711yescharacteristic
211C2×C47#191411yescharacteristic
212D47—162no—
213C2×#131—21no—
214#532C2×D4711yescharacteristic
215C2×#127—471no—
216#493—941no—
217#486—941no—
218#459—942no—
219#447—474no—
220#436—21no—
221#435—22no—
222C8×D4—21no—
223#431—471no—
224#423—471no—
225C2×C4⋊C8—471no—
226C2×C4⋊C8C2×D4711yescharacteristic
227C2×#100—941no—
228C2×#100—941no—
229#408—21no—
230C4×M16C2×D4711yescharacteristic
231#332—941no—
232#331—941no—
233#331—941no—
234#331—941no—
235C47#927311yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C22×D4—472no—
238C22×D4—941no—
239C22×D4—941no—
240C2×D8—944no—
241#131—42no—
242#131—42no—
243C2×M16—22no—
244C2×M16—941no—
245C2×M16C22⋊D4711yescharacteristic
246C2×M16C22⋊D4711yescharacteristic
247C22×C8—22no—
248C22×C8—21no—
249C22×C8—941no—
250C4⋊Q8D18811yescharacteristic
251#127—941no—
252#127—941no—
253#127—941no—
254#127—941no—
255#127D18811yescharacteristic
256#124—21no—
257#121—21no—
258#121C4×D4712yes—
259C4×D4—21no—
260C4×D4C22⋊D4712yes—
261C2×C42C22×D4711yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—941no—
264C4⋊C8—941no—
265C4⋊C8—941no—
266C4⋊C8—941no—
267C4⋊C8—941no—
268C4⋊C8C22⋊D4711yescharacteristic
269C4⋊C8C22⋊D4711yescharacteristic
270#102—944no—
271#100—1882no—
272#100—1882no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—472no—
276#97—21no—
277C4×C8—21no—
278C4×C8—941no—
279C24—942no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—942no—
289C2×D4—942no—
290C2×D4—3761no—
291C2×D4—3761no—
292C2×D4—3762no—
293C2×D4—1881no—
294C2×D4—1881no—
295C2×D4—1881no—
296C2×D4—1881no—
297C2×D4—1881no—
298C2×D4—1881no—
299C2×D4—1881no—
300C2×D4—1881no—
301C2×D4(C2×D47)⋊C412yes—
302C22×C4—22no—
303C22×C4C2×C4×D4711yescharacteristic
304C22×C4#1062535011yescharacteristic
305C22×C4C2×C22⋊D4711yescharacteristic
306D8—3764no—
307M16—21no—
308M16—21no—
309M16—21no—
310M16—44no—
311M16—1881no—
312M16—1881no—
313M16#1062533412yescomplemented
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—44no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—942no—
322C2×C8—3761no—
323C2×C8—1881no—
324C2×C8—1881no—
325C4⋊C4—21no—
326C4⋊C4(C2×D47)⋊C412yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C42C2×D18811yescharacteristic
330C42#1062535011yescharacteristic
331C42C2×C22⋊D4711yescharacteristic
332C42C2×C22⋊D4711yescharacteristic
333C23—22no—
334C23—3762no—
335C23—3762no—
336C23—1882no—
337C23—1882no—
338C23—1882no—
339C23C4×C22⋊D4711yescharacteristic
340Q8—41no—
341Q8—41no—
342D4—42no—
343D4—42no—
344D4—42no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—3762no—
350D4—3762no—
351D4—3761no—
352D4—3761no—
353D4—3761no—
354D4—3761no—
355D4—3761no—
356D4—3761no—
357D4—1884no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4C2×(C2×D47)⋊C411yescharacteristic
365C2×C4C4×C22⋊D4711yescharacteristic
366C2×C4#1309399311yescharacteristic
367C2×C4#1309399811yescharacteristic
368C2×C4#1309399811yescharacteristic
369C2×C4#1309400711yescharacteristic
370C2×C4C2×#1062533411yescharacteristic
371C2×C4#1309404411yescharacteristic
372C2×C4#1309404911yescharacteristic
373C2×C4#1309405011yescharacteristic
374C8—22no—
375C8—41no—
376C8—41no—
377C8—41no—
378C8—3761no—
379C8—3761no—
380C8—1882no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—3762no—
387C22—3762no—
388C22—3762no—
389C22—7522no—
390C22—7522no—
391C22#1568900211yescharacteristic
392C22C4×#1062533411yescharacteristic
393C22#1568923311yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4#1568925711yescharacteristic
399C4#1568926811yescharacteristic
400C4#1568931811yescharacteristic
401C4#1568933811yescharacteristic
402C4#1568938611yescharacteristic
403C4#1568939311yescharacteristic
404C2—21no—
405C2—21no—
406C2—82no—
407C2—7522no—
408C2#2024373611yescharacteristic
409C2#2024389611yescharacteristic
410C2#2024393511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2935C2
socleC22×C47#386
prime radical#20254303C2
cosocleC2×C42×C47C23
hypercenter#2935D47
hypocenterC47#9273
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20244228C2semidirect
3. #20242065C2semidirect
4. C2×#15688866C2semidirect
5. #20241285C2semidirect
6. #20241239C2semidirect
7. C47×#2935C2semidirect
104. M16×C47D8semidirect
200. #2935D47semidirect
235. C47#9273semidirect
313. M16#10625334semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202543032
#202442282
#202420652
C2×#156888662
#202412852
#202412392
C47×#29352
#927347